Level set

A level set of a function $f$ is a set of the form $\{x:f(x)=c\}$.

A level curve is the planar level set of a scalar function. Where the gradient does not vanish, the implicit function theorem makes it locally a smooth curve. At a critical point, branches may cross, narrow into a cusp or shrink to a point, so a plot should be interpreted with the local critical-point analysis.

A level set at a regular value is a smooth submanifold. By the regular level set theorem, its codimension is the dimension of the target. For determinant-one real symmetric matrices, $D(\det {)}_B(C)=\mathrm{tr}(B^{-1}C)$ is nonzero because $C=B$ gives $n$, so the constraint has codimension one.

A superlevel set of a real-valued function $f$ is a set of the form $\{x:f(x)\ge c\}$ or $\{x:f(x)>c\}$.